Founded by husband-and-wife team Tom Costello and Anna Patterson, Cuil aims to rank the relevancy search results by content analysis rather than by popularity.

Cuil, a new search engine, opened for business on Monday, boasting an index of 120 billion Web pages, “three times more than any other search engine.”

Google (NSDQ: GOOG), as if pre-briefed on today’s announcement, on Friday said that its index had reached 1 trillion URLs, though not all of them lead to unique Web pages.

Founded by husband-and-wife team Tom Costello and Anna Patterson, a search-engine researcher from Stanford University and a Google technical lead, respectively, Cuil aims to rank the relevance of search results by content analysis rather than by popularity. It’s an obvious swipe at Google, which treats Web links as popular votes in weighing Web page relevance for a given query.

“Our team approaches search differently,” said Patterson, president and COO of Cuil, in a statement. “By leveraging our expertise in search architecture and relevance methods, we’ve built a more efficient yet richer search engine from the ground up. The Internet has grown and we think it’s time search did, too.”

For Cuil, pronounced “Cool” rather than “Quill,” there’s a separation between search and surveillance. Whereas Google records information about its users and their searches to improve the user experience and to deliver more relevant search results and ads, Cuil remembers nothing.

Cuil’s privacy policy actually promises privacy: “When you search with Cuil, we do not collect any personally identifiable information, period. We have no idea who sends queries: not by name, not by IP address, and not by cookies (more on this later). Your search history is your business, not ours.”

Cuil is not the first search engine to offer search privacy; Ixquick has done so since June 2006. Ixquick, as a metasearch engine, gets results from other search engines. Cuil, however, offers new search technology and a column-based search results interface.

It remains to be seen whether computer users care about privacy enough to alter their search habits.

In terms of performance, Cuil is responsive and looks good, at least when the service is up. The search startup was down at least twice on Monday. “Due to overwhelming interest, our Cuil servers are running a bit hot right now,” a Cuil error message said just before noon Pacific time on Monday. “The search engine is momentarily unavailable as we add more capacity.”

But Cuil leaves something to be desired in terms of the relevance of the images it places beside search results. “We do our best to take images from Web pages that accurately reflect the content of the Web site,” Cuil’s FAQs document explains. “Many Web sites are full of images, so we use advanced algorithms to determine the best image to show the user.”

Advanced though they may be, Cuil’s algorithms aren’t yet accurate when it comes to placing images with links to related content. An ego-search for my name, for example, used a CNNMoney.com graphic to link to a blog that published a review of a book I wrote.

Google News has had similar issues, associating unrelated text and images. But such inaccuracies seem to be increasingly rare.

Though Cuil may aspire to challenge Google, it has some basic service and accuracy issues to deal with first. After that, it can join Exalead, Hakia, Mahalo, Powerset, and Wikia in their quests to challenge the big-league search engines.